Complementary Error Function Calculator

The complementary error function is defined as erfc(x) = 1 − erf(x), where erf(x) is the Gauss error function. It equals (2/√π) times the integral of exp(−t²) from x to infinity. The result always lies between 0 and 2 for real inputs.

They always sum to exactly 1: erf(x) + erfc(x) = 1 for all real x. This means erfc(x) = 1 − erf(x) and erf(x) = 1 − erfc(x). The calculator displays both values so you can verify this identity directly.

For real inputs, erfc(x) ranges from 2 (as x → −∞) to 0 (as x → +∞). At x = 0, erfc(0) = 1 exactly. It is a strictly decreasing function across all real numbers.

The error function is erf(x) = (2/√π) ∫₀ˣ exp(−t²) dt. It is not expressible in terms of elementary functions, so numerical approximation methods — such as Taylor series or rational approximations — are used to compute it.

The erfc function is directly related to the tail probability of the normal (Gaussian) distribution. The probability that a standard normal variable exceeds a threshold z can be expressed as erfc(z/√2)/2, making it central to hypothesis testing and signal detection.

Yes. For negative x, erfc(x) = 1 − erf(x) and since erf is an odd function (erf(−x) = −erf(x)), erfc(−x) = 1 + erf(x). This means erfc values for negative inputs are greater than 1.

The inverse complementary error function, erfc⁻¹(y), returns the value x such that erfc(x) = y. It is defined for y ∈ (0, 2). This calculator focuses on the forward erfc(x) computation; for the inverse, you would apply a root-finding method to the erfc formula.

This calculator uses a high-accuracy rational approximation of erf(x) (Horner-form coefficients accurate to 15+ significant digits) and derives erfc(x) = 1 − erf(x). You can select up to 15 decimal places of displayed precision.